-
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) - c. I' L / ,7-
NASA CR-I208!j4
' \ 0
TOPICAL REPORT
THERMAL CONDUCTIVITY AND ELECTRICAL RESISTIVITV OF POROUS
MATERIAL
BY
J. C. Y. KOH, The Boeing Company
ANTHONY FORTINI, NASA-Lewis Research Center
PREPARED FOR
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION OCTOBER 1971
THE
CONTRACT NAS 3-12012
i i
I . ?
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NASA CR-120854
TOPICAL REPORT
THERMAL CONDUCTIVITY AND ELECTRICAL RESISTIVITY OF POROUS
MATERIAL
BY
J. C. Y. KOH, The Boeing Company
ANTHONY FORTI N I, NASA-Lewis Research Center
PREPARED FGR
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION OCTOBER 197?
CONTRACT NAS 3-12012
THE B ~ ! I / W & C O M P A N Y AEROSPACE GROUP
SEATTLE, WASHINGTON
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Them1 conductivity a d electrical r e s i s t i v i t y of
pornus mtcrials, i n c l u d i w 3ObL s t a i n l e s s steel
Rigimesh, 304L stainless steel s in te red spher ica l powders, and
QFHC s in te red spherical powders a t d i f f e r e n t porosities
and temperatures are reported and correlated. It was found that t h
e t h e m 1 conductivity and electrical r e s i s t i v i t y can
be related t o the sol id mterial properties and the porosity of t
h e porous IIlatrix regardless of the rmtrix structure. It was also
found that t h e Wiedermann-Franz-Larenz re la t ionship is v a l i
d for t h e porous lmterials under considerstton. For high
conductivity loatcrlals, t h e Lorenz constant and the l a t t i ce
component of conductivity depend on t h e loaterial. and are
independent of t h e porosity. For l o w conductivity, t h e
lattice component depends on the porosi ty as w e l l .
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TABLE OF C0R"JWT
Page no.
ABSTRACT
1.0 I r n O r n O r ?
2 .o MATERIALS STUDIED
3.0 RESULTS AKD DISCLPiSI9IOS
3.1 THERMAL COI9Du;?pMTY
il
1
1
2
2
3.2 ELECTRICAL RESISTIVII?! 3
3.5 RELATION BETwmv trHmMAL C O r n r n N I T Y m 6 ELECTRICAL
RESISTIVITY
4.1 PREDICTION OF W R O E MATERIAL CO!VDfX)TIVITY FROM 9 SOT;11)
COIODfx3TTVflcy DATA
4.2 PIZEDICL'ION CF POROtE PWBRIAL CO?lDWTIVITY F R O M 9
ELEZTRICAI. RESLSIT\IITT DATA
5.0 c0mus10R3 Am) F4EcoMmTxon~ 10
5 . 1 COr?CLtGIOIQs 10
5.2 RECOMMENDA!LTOBS !?
6.0 HEFEREm 11
I O O M E B C U W
TABLES
FXGWS
A P Z X f D I X A - BU)rlERICALEXAMrmE
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LIST OF TABLES
I
II
111
Iv
v
VI
VI1
Details of Tested Samples 1L
Thermal ConductivTty, Electricel Pesintivity and 15 Dex5ved
Lorenz Amction for Porous ?94L Stainless Steel
(a) Rigimesh
(b) S i n t e n d Pwders 16
Temperature Coefficients 18
Mmnsimless Condv.cti\'ty an& Resistitrlty of 9 4 L
Dimensionless Conductivity and k ~ i s t i v i t y of
20 Stainless Steel Sintered Rwders
21 Copper Sir,tered Pjwders
i V
-
LIST OF FIGUIPES
1.
3 c.
3.
4.
5.
6.
7.
a.
9.
10 .
11.
12.
Thermal Conductivity of Porous 304 and 304L Stainless Steel
Thermal Conductivity of Capper Sintered Pawders
Electrical Resistivity of Porous 34L Stainless Steel
Electrical Resistivity Olf Copper Sintemd Powders
Comparison of Ekperimental Data with Comlations of Thermal
Conductiuity of Porous hterlrrls
Thermal conductivity vs. Temperatllm?/Edesistivity of Porous
304L Stainless Steel RigLmesh
Thermal c a u c t i v i t y vs . Tempereture/ResistivIty of 304L
Stainless Steel Sintered Puwders
!rhermsl Conductivity vs Tsmpemt~JIpesistivity of OFIiC Sintered
patders
Correlation of Thermal Conductivity, Electrical Resistivitq
Porosity, and Teqpenature of 304L Stsin- less Steel Rigimesh
comT!latioa of Thermal conductivity, Electrical Eaasistlvity,
Porosity, and Te-mture of 304L Stainless Steel Powders
C o r r e l a t i o n of !Chermsl Conductivity, Electrical bslst
ivity, Porosity, and ‘pempemtum of 304L Stainless Steel Rlgtmesh
and Pawders
page
22
23
24
25
26
rl
28
29
30
31
32
33
V
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1.0 IN!EIODUX!IOFI
Them1 conductivity of porous nraterial is an important property
i n deternining t h e temperature dis t r ibut ion of coolant and
porous s t ructure i n transpiration cooling. Due to the i r
regular i ty of t h e mlcrostructures, confident theoret ical
calculation of thermal conductivity of porous mterlals is rather d
i f f i cu l t if not impossible. such as parallel cylinders,
laminates i n series, spheres dispersed i n a conduct- ing medium,
e tc . Even with a w e l l defined mlcrostructure, the problem
remains complex due t o the existence of the interface resistance.
Therefore, with the exception of parallel cylinders, a
sed-empirical approach Is the only pract ical way of predicting the
t h e m 1 conductivity of porous materials. This approach was used
by Groetenhuis, Puwell and !&e (Reference 1) who measured the
thermal conductivity and e lec t r ica l r e s i s t i v i ty of
sintered p d e r bronze at d i f f e r e n t porosities ( from 2OoC
t o 2oo0c. They found tha t a l l the data on thermal conductivity
X and electrical r e s i s t i v i ty a different temperatures T
may be represented by a straight l i n e h = 2.43 x loob 3 + 2.1
Independent of porosities. They a l so suggested the use of ' = 1 -
2.1 c for thermal conductivity calculations where Lo is the therm1
conductivity of solid mterial . for t h e thermal conductivity ai
porous materials as follows: Ta = * n is a constant depending on
the microstructures. expressions for other microstructures and
other rsaterials remains unknm.. The purpose of the present study
is t o provide experimental data of themEd conductivity and e l ec
t r i ca l r e s i s t i v i ty of different parous materials a t a
temperature range from 100% t o lOOO% and t o develop
semi-empirical equations from these data.
Existing prediction methods are based on cer ta in s iapl i f
icat ions
Recently, Aivazov and Donrashabv (Reference 2) derived an
expression
The val idi ty of t h e s t
The materials studied are (1) porous 304L stainless steel
Rigimsh, (2) 304L stainless steel eintered spherical-powder, an8
(3) oxygen free high conductivity copper sintered spherical-powder.
Rigimesh, stainless powders and copper powders respectively in t h
i s report. Three different porositiss of each mterial are
investigated. tested samples is shown i n Table I. "Flar" applies t
o either the heat f law direct ion i n the case of conductivity
measurepnents or the current flow direction i n the case of
electrical r e s i s t i v i ty measurements. For the solid
materials and sintered powder samples, the materials are Isotropic
and the f l a w is always aloag the "1 " direction desigarrted
under column 4. A dc+iailed characterization of the porous
materials may be found i n References 3, 4 an8 5 .
For convenience, these materials w i l l be called
A summary of the In t he last column of the Table, the term
The thermal ccmductivlts and electrical r e s i s t i v i ty of
a l l samples were measured by Dynatech Corporatian. Detailed
description of the t e s t e was reported i n Rcierencea I and
6.
1
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3 - 0 €?ESuI;Ts AHD DISCUSSIOlU3
3.1.1 Porous 30bL Stsinleas S3eal. The them1 conductivity of
porous 304L stainless steel at various porositids and temperatures
is sham both i n Table I1 and Figure 1. The effect of therms1
expansiYn on length and area of the sample has not been included.
Available data i n the literature for solid 304 and 304L stainless
steel are a180 shown in Figure 1 for comparison. Reason- able
sgreemont is found between the published results and the present
data. For the 841ue material a t various porosities, no va l id
published data are avail- able for comparison.
Figure 1 shows that a t low porosity ( f & O . l ) , the
thermal conductivity of the Riglmesh is significantly higher than
that of the sintered powder. This may be due t o the experimental
error of the Riglmesh.
3.1.2 Porous C o p r . h e them1 conductivlty of copper at
various porosities and temperatures is shown in Table If1 and
Figure 2. results of the conductivity of solid copper are a l s o
sham in the Figure for comparison. equation given i n Reference
12.
Seveml published
The lines for pure and impure copper were computed by the
following
where:
1330 w m o l Isg1 for impure Cu
. T = Temperature (k)
The present data for the so l i4 copper may be represented by
the follarlng squation:for the temperature range from 300 K t o
lo00 K.
b
Clearly, Figure 2 ehowe that the conductivity of copper is a
significant function of' impurity. If it is assumed that the resu l
t s for pure coppr, .(# Cu are reliable, then the valws of
conductivity for .99989 Cu an8 .m Cu appear t o be too law. In
general, it nray be stated that within the temperature raw from 300
K to 1300 K, the thermal conductivity of porous capper decresses
linearly aa the temperature increataes, t h e m 1 conductivity
decreases with tmperrctum a t a rate signif icant ly higher than at
any other poro8itiee. for the sample at ,2096 porosity.
Cu and
19otice that for the porosity of ,2096, the
This Indicate6 the possible experimental e r ro r
2
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3 02 ELECTRICAL RESISTNrn
3.2.1 3 O b L Stainless Steel. The e lec t r i ca l r e s i s t
i v i ty of porous 304L stainless steel a8 a function of temperatme
a t ,clrious porosity le shown both i n Table I1 and Figure 3.
expansion. f o r comparison. with those of Reference 11 quite w e l
l . discrepancy is significant. e lec t r ica l r e s i e t iv i ty
a t a temperature higher than W K . is not found in the present
data. It is not found i n the conduct iAy inform- t ion either
(Figure 1). and thermal conductivity, it is concluded tha' the
rapid increase of electrical res i s t iv i ty a t the temperature
beyond 900% as shown i n Reference 11 may not be v a l i d .
Again, t h e data have not been-corrected f o r thermal
Published data for the solid mterial are a l so shown In the
Figure
For a temperature up t o about W@K, the present results agree
Huwever, for higher temperature the
This i.)ri.?nomnon Data f r o m Reference 11 shows a rapid
Increase of
In view of the s i m i l a r i t y between e l e c t r i c d
conductivity
3.2.2 Sintered Powder Cop- r. The electrical r e s i s t i v i
ty ai copper a t various porosities and temperatures are sham I n
Table 111 aad Figure 4. Resistivity information for solid copper as
a function of temperature published in Reference 11 1s. also shun
in t h e Figure for comparison. Agreement between the present
results and those of Reference 11 on solid copper resistivity is
quite good.
3.3 !rEMPEIzATuRE EFFECTS 't
The effect of temperatm on both t h e t h e m 1 conductivity and
electrical r e s i s t i v i ty is represented in the form of
temperature coefficient i n Table IV. The temperature coefficient
is computed from the experimental data given i n Tables I1 and I11
bythe follawing equations:
Where the subscript indicatesWra temperature i n degrees C. Tho
temperature coefficients for bronze p d e m are also s h m In Table
IV, for comparison. Table IV shows that for the sintered powders
the temperature coefficients o( and 4 a t 21$ poroeity ie signif
icant ly different from the other goroelties. As pointed out
pruviously, this raay be due t o the experisrrsntal error ab the
eamle a t 21s porosity. With this in mind, Table N shows tha t the
temperature coefficients are practically constent independent of
porosity. Thus, the t h e m 1 conductivity an4 electrical m s l e t
l v i t y at any temperature may be comgutsd by the iollowing
eqmtlons:
3
-
where:
- - A T - I t o ( ( T = r o o ) I O 0
( 5 )
.000544 stainless steel B " .GO329 copper
3.4 DIBIOIUJBS CORDU2IVITY AND RESfsTIVITY
Using the thermal conductivity a d electrical resistivity of
solid material as a reference, the dlmsnsionlsss them1
conductivity,
as a function of temperature for different less electrical
resistivity, porous lnaterial were cwputed from Tablee fI am3 111.
The comrputod results are listed in Tables V, VI, and V I I
respectively for the stainless steel Riglmesh, the stainless steel
powder, and the copper pders. An inspection of Tables V t o V I 1
reveals t b t the dimenrrionless thermal conductivity a d
electrical resistivity are essentially indeprdeat of temperature.
This is consistent with the expressione of equations (5) and (6 ) .
'pbus, an average tralue for each =torial and each porosity may be
use4 in the 8ubsequent dlscussisns. These average values a m shown
i n Figure 5. The solid lines in Figure 5 are represented by a
correlation discussed in the following section.
A A 0
and the dimension- ' O
3-
3.4.1 Corralation of Mmsnsionless Conductivity a d Resietivit--.
Mfferent c o r r e h t i a n eqmtians have been published in the
literatures tRef 2, 13-18). It ha8 Men found that the present
experimental data can I* bet represented by either ol~t of the
folllrrwing two eqmtions:
ices
-
Equation (7) wa8 derived f'rom a general form of the thermal
conductivity of a mixture (Reference8 13 and 3.4). material having
different regular pore configurations (Reference 2). ao, a1 and n
are constants depending on the microstructure of the material and
must be determined from experimental data.
Equation (8) was deduced from a poroua
Since equation [8) is significantly simpler than equation ( T )
, it w i l l be used i n the p~esent correlation. of and 5 t o
yield:
Equation (8) m y be solved for n i n terms
Xa
Thus, using experimental &+a, n can be found for each pair
of values of c and -ee x Rigimsh and sintered staitless steel at
about 9.a porosity for the reason that there may be e q p r i r e n
t a l errors , it w a s found that t h e value of n varies from 8.1
to U.2 and the mean value for n is 10.2. of 1.C: will be used for n
and the dimensionless conductivity is given by
. Osiw the results i n Tables V, VI, and V I 1 and excluding
the
Therefore, a round n&r
Similarly, the correlation of
-c, go 1 - e -
J I + I I p
(10)
e lec t r ica l Yesistivity is given by:
(11)
Eqw t ions (10) and (U) are s h m as solid lines in Figure 5.
Considering the ccmplexity of the problem and the s ignif icant
differences i n ndcrostructms from Riglmeeh t o sintered powders,
It is concluded from F i w 5 that the correlation r~ satisfactory.
'Whether or not t h l e correlation is satisfactory f o r other
microstructures, such as foam metal, felt metal, lamllloy, poroloy,
ctc., remains t o be substantiated.
3.4.2 Reference 1 that for rintercd nmtrlcei, the eHgerimsnta1
data 011 thermal con- ductlvlty can be b a t correlated by
Other C0rrclat; Iana and lExps rimbntal Oata. It was ahawn i
n
sinple s+,rzrl@t line as follaws:
x A 0 - - - I - 2.1 f
5
-
A n extensive revlev of 11ttratu-e on them1 coduc t iv l ty of
porous materials perQonaed i n Reference 19 shave that equation (a)
indeed represents the avail- able experimental data well. and the d
e l of parauCl cylinders, experimental data are also sham i n the
Figure for canrparlson. Hovever, for clarity, the experiiaental
da+a fw sintered powders compiled in References 1 and 19 are not s
h u n i n the figure.
Figure 6 shows t h a t the equation he= I + 105 correlates a l l
the experimental data very well. would over estl tmte the
conductivity while the carrclation& = 1 -
This correlation, together w i t h equation (10) = 1 - f are
shown in Figure 6. S s
On the other hand, the parallel cylinder model,+*= I - f would
under estimte the conductivity when the porosity is larger
?lgurc 6 shuus also that t h e conductivity far Foameta1 ardl
Feltmetal a+, hi@ porosity ( f 3 b2$) can ?)e represented by
equation (10) . for these materials at low porosities are avaiLable
t o subtan t ia te t h e correlation.
Bo experilaen*d data
3.5.1 Relath Ibtabllehed i n LlteratuTes. hblished literatures
(Reference 23) shaw that vlthln the temperature range where there
is no -tic transformtion the them1 conductivity and electrical r e
s i s t i v i t y of bolld materials are appraxlmktely related by
the following Wledernzmn-Pranz- Lorem equation
where L = Lareneccmstant
LT - = Blectronic ccmpment of therm1 conductivity b - Lattice
cmtponent of t h e m 1 conductivity f
Values of L and b ror C?+"f'erent ranterials can be found i n
Reference 23.
T h V s . 7 of Present hta. Ulslng the data in Bablee I1 and
111, T
3.5.2 thc functional rclatlonahip between x and 7 for the
present data is shown in Piguree 7, 8, and 9 for the etalnltse
Plglmesh, stainlees parWacrs, and ceper parodcrs respectively. that
d ie t iac t straight l h e e could be drawn thrOugh the data for
each porosity. Abo, the slope I s e8oentlally the mame between
different llors and Independen+, ob parobity. Themfore, for the t
.Inless steel material, the Lorsne constant in equstion (13) IS a
conatant i n d w r " v t o0 porosity a i l e the lattice
colsrponaant of conbuctl,rity b depldft an t?. i**.,. For t he
copper nater ia l , Figure 9 show8 that a l l *&e data a t ' -
' \I?-Y r a p e e n t e d by a eingle line lndqmmlent ob pawltjr.
thing BL 1.' t *tho& the l ine represents best the exporixentsl
i e a8 P0114~oe*
For the stalnlear steel matm-ials, Flgurcs 7 and 8 shw
-
2.307 X loo8 3. 18.6 (Copper Powders) (14) t I n Reference 1 it
is shown that the them1 conductivity and electrical r e s i s t i v
i ty of bronze puwders can be represented by the followtag re lat
ion independent of porosity.
Material
Copger a t 100 c
kmze a t S0”c Stainless Steel af l 0 0 O C
So far, the available experlnental bata for porous agterlals
show that:
Conductivity
x xe xm - A, Porosity w-m’ 1 ~ 1 (LT/f) (b) x Source
3009 163 Wt 18.6 . 114 Resent . 3% 15.1 13 2.1 . 14 Ref. 1 385
4.3 2.65 1.65 .304 Present
(1) h e Wiedermnnn-hnz-brenz re la t ion given by equation (13)
is valid. (2) The Lorenz constant L depends on the bind &
leaterial. and independent of porosity. of material for bronze and
copper. stainless steel.
( 3 ) The Lattice component of conductivity b depends on t h e
kind It depeds an the porcmity as well for the
To explain the different dependency of b on porosity, the
following infarmation on conductivities of porous copper, bronze,
and stainless 8-1 Mgimesh at 50% - 100% is provided.
The above table shows that for porous bronze and copper, the
lattice component oi t h e e conductivity is relatively unimportant
(up t o 14s of total con- ductivity) while for porous stainless
steel Rlglxssh, the lattice campanent of conduction (38$) ir ;
almost as impartant as the electronic component UP conduction.
Therefore, it is appropriate t o postulate that within the
temperature range where there i r no =-tic transfannation the
lattice ccmponent of conduction depends on the parosity for all the
matariala. But, due t o the l f m a i t in experimental accuracy,
the effect of poroelty on the lattice coqpment of conductivity
(i.ee9 b in equat1c.n 13) can only be found in lcrv u d u c t i v i
t y m t e r i a l e where the lattice conductivity i e important
an8 can not b8 foW !,a high conductivity laaterials where the l a t
t i c e conductivity is r0Irrtivd.y
7
-
unimportant. and hig!!er conductivity rmterlals, the thermal
conductivity and r e s i s t i v i t y is related by a single line
of eqmtion (13) v i t h e single slope L and a single intercept b
independent of porosity. ductivity materiale, the them1
conductivity and e l e c t r i c a l resistivl%y is related by a
set of straight lines.
For engineering applications, i t m y & stated that for
bronze
For stainless steel and low con-
T 3.503 Carrelation of 1 v8. 7 for 3&L Stainless st@el
RigiElesh, me experilsental data on therm1 conductivity and
electrical r e s i s t i v i t y a t different temperatures and
porosities are correlated to a single equation by the following
postulatim:
(1) me SI- L i n equation (1.3) is a canstant idependent CYP
porosity.
(2) me lattice ccmponent of conductivity b is a linear function
of porosity, i.e.,
b
~hus, equation (13) am be rewritten as
x = T L- r + co - c1 s The constante, '3, Cc, and C 1 are found
by the following steps:
(1) For each poroeity, dt!termlrn? the best slope L Sy a least
square mcthdl.
(2) All the values ai L found in Step (1) are added a d divided
by the number of' porosities. The result I s ta.ken as the best
value for L.
(3) bi-the best value of L in equstfon (16) and the experinrntal
data of X , T , a d f , the best values of Co and C1 are determined
by the
least square method.
The resulting equation as found by t.ks foregoing calctiktions
for the 3m1 stainless steel Rigimesh is ae followe:
+ 9.236 f -0 = 2.292 x 10 T - t + 5.309 (Rigimesh) Figure 10
shows a campax?son bqtweea the experbmntal &ta snd the
correlation. With the exceptions of two dsta points the), b v i a *
a frcm the c m k t i c n eqwitticm by 6.6$, over TO$ of ~e da* fall
within 3$ of the correlation given by equation (17).
T 3.5.4 c ~ ~ ~ ~ ~ h t i a af >. ~ 8 . for 304~ stainless
Stee l P d e r e . ma B a m e proce8uree gutllnad i n that
foregoing section have teen employed t o correlate the date for the
304L stainless a t e e l powders. bslaw . The resulting equation is
3 h m
8
-
A camparison of the correlation l i n e an8 the experimental
data is shown i n Figure 11. is 9; over '755 of the data are within
2$ of the correlation. f m t h e powders a p r t o be better than
that for the Rigimesh.
The lllsxitrci deviation between the l i ne and the experimental
data Thus correlatioii
corre'-atiOn V s . for 304L Stainless Steel Riginesh and When
the cnta for both the Rigimesh and powders art considered together,
- 3.55
Pawders. t he following !m-relation is obtainerl:
-8 T 1
X + 11.18 f = 2.16 x io - + 5.995 This correlation is Shawn in
Figure 12 together with the experimental data. The maximum
deviation 'between the correlatiort iiue arid the experimental data
is g$ f o r the Rigimesh at -0% poroeity. of the value given by
equation (19). this correlation is deerned satisfactory.
Over TO$ of the data fa l l within In view CYP the complexity af
the problem,
In general, the thermal conductivity and e lec t r i ca l r e s
i s t i v i t y of a sol id material can be found i n the
literatures. This infornmrtlon lnay be used t o estimte the therm1
conductivity of t h e lnaterial a t different porosity and
temperature
When the t h e m 1 conductivity of' so l id laaterial as a
function CFP temperature is ham, the therm1 conductivity of porous
material can be computed use of equation (10). When the thermal
conductivity of solid maaterial is known only at a certain
temperature, the them1 conductivity of porous material a t any
ozher temperdture m y be found by first computing the solid
laaterial conductivity from an equation sindlar t o equation ( 5 )
and then obtain the answer by use of equation (10).
When the electrical r e s i s t i v i t y of a solid material a8
a function of temperature is given, the thenaal conductivity of p m
u 8 miterial can be found by two steps:
(1) Determine thr them1 conductivity of solid materlnl as a
function of temparatur, by u88 W eq\loation (13) w i t h
f%pproprlatc constant8 L and b.
9
-
( 2 ) CampUte tkbe thermal conductivity of poms material by use
of equation (10) . When the electrical resis%iVity of 8 solid
material is known at a temperature only, the thermal conductivity
at the same buperature can be foimt! by use of equation (13).
ductivity at eny other .t;eqx?rature. i.Yna&y, the t h e m 1
conductivity of the material at a specific porosity can be computed
by -me of equation (10).
mu8tdarA (5) can then be used to determine the s o l i d
con-
A nuarerical exan&le of the caqputstlontrl procedure as
outlined above is presented i n Appendix A.
Thennal conductivity ami electrical ?esist!.vity of stainless
steel Rigimsh, sTain3ess steel powders and copper sintemd
sphx-tcal-gcrwders at different porosities have been measu=d for a
temperature rarqp fmn 100°C us to 1000°C. &ita have been
analyzed and correlated. existing data for ather porous materials.
cmclusiaas be drsam.
The cormlatioas were tested using &wed on t h i s study, the
follaKing
(1) For sintered pcrwder and Mgimesh, the dimensionless thermal
coslductivity
resistivity I s given by - fo - - - e . '~?ae cornhtion also
fits t ~ a e exlsting data on thermal. conductfvity crf f m t a l ,
feltmetal, an& non- swricsl sintered powders.
(2) tk ' t e Q C ~ ~ W h e r e there 18 RO lnagnetfc
tranSfmtiCXl, the thermal conductivfty of pOrCKl8 metals I s
x-zlate!d to t h e ekctr ica l resistivity tlILd temperature by the
W¶.~rmanm-Franz-brerz equation:
x = - + b. ?or a high conductivity matertal, such as bronze and
P
copper, when! the lattice crsrponent of conduction is relatively
unimportant the slope L emd the intercept b axe a function of
matellal but indapndent of poroeity. law conductivity matertal such
as stainless s*el w h e r e +h h%tice componant of conductivity is
sbno6t as important 86 the electronic component of cmductivity.
Havever, *As intercept &epOnits GIY the pomity 88 w e l l
as
(3) For Rigimsh and sintered pawdar structures, the thermal
conduclivity of poraus uiaterlals can ?E ccrmputed fnw the
lnfofnration of' soUd matertal conductlvtty or so l id material
electrical resistivity.
5.2
'Phfs stud$ bes been U t e d to ths porous Rtgirpesh and sintemd
wwder structure. It i e unknawn wbmtbbr or nut the canclurpiopls
are vaUd for other microstructure of po2uws n 1 8 t e r l m l r ,
mch ae f'e1-tal, at lcrw porosity, foaPztsl at low poroeity,
lmillq, porOragt, ate. It l e , therefore, eu@mted that atudy be
e-utandcd t o such diffwmt mlcroetructume c-ly ured In engineering
p m e s e s .
-
1.
2.
3 .
4.
5 .
6 .
7-
8.
9.
10.
11 . 12.
13
14.
15
P. Grootenhus, R. W. Powell, and R. P. "ye, "Therm1 and Electr
ical Con- ductivity of Porous Metal bkde by Powder Metallurgy
Methods, I1 Proc. phys. SOC., B 65 (19%) 5W-511.
M. I. Aivazov and I. A. Donrashnev, "Inflwnce of Porosity on the
Con- ductivity of Hot-Pressed Titanium-Nitride Specimens," Ins t i
t u t e for New Chemical Problem, Acadeqy of Sciences of the USSR.
Translated from Poroshkovaya Metallurgiya No. 9, (69), pp 51-54,
September 1968. R. E. Regen, "Characterization of Porous Matrices
for Transpiration Cooled Structures ,'I NASA CR-MB ( I l g O )
.
G . Friedmm, "Fsbrication, Characterization and Them1
Conductivity of Porcus Copper and Stainless Steel hterials," roAsA
CR 72755 (lw).
R. E. Regan, "Characterization of Porous Powder &tal
&trices for Transpiration Cooled Structures," M A CR -994 (191)
.
R. P. Tye, "An Experimental Investigation of the Thermal
Conductivity and Electrical Resistivity of !Three Porous 304L
Stainless Steel "Rigimesh" Material t o l3oO%'', NASA (3-72710
(190).
Aerospace Structural Metals Handbook, ASD-rn-63-741, Vol. I,
1963. Force Materials Laboratory, Wright Patterson A i r Force
Base, Ohio.
A i r
Argonne National Laboratory, ANL-5914, September 1958, p.
64.
ASTM Special Technical Publication, 100. 296, p. 67.
Metals Handbook, 8 t h Edition, Vol. 1, Properties and
Selection, American Society for ~etals, page 122.
Y. So Touloakian, "Themnophysical Froperties of High
Temperatures Solid lbterials," Vol. I, 111, MscHlian C-, Aew York,
(1967).
A. Cezairliyan anb Y. S. Toulouktan, "Prediction of Themml
Conductiv5ty of Metals," 3rd Conference on Thermal Conductivity,
Oak Ridge National Laboratory, Aok Ridge, Tennessee (1963). pp.
3-19.
D. A. G. Bruggemn, "Dielec5ric Constant and Conductivity of
Mixtures of Isotropic hhtcriaJ.s," Annulen Phy~ik, Vol. 24, 1935,
pp. 636-679.
I*
A . E, Pcrwers, "FunBamentala of Them1 Conductiv: t y et Hi.@
Temperatures Knolls A t d c Puwer Laboratory, Report No.
KAPL-22',&3, Ap?:ll 7, 1961. V. V . Skorokhod, "Electrical
Conductivity, M a d u l ~ s of Elas t ic i ty Mnd Viscosity
Coefficients of Porous Bodies,'' Powder Mezallurgy, 1963 no. 12, p.
188-2000
1.1
-
16.
17
18 .
19.
20.
21.
22.
23
A. V. Kuikov, A. T. ShaBhkov, L. L. Vasiliev, and Yu. E.
Fraiman, "Thermal Conductivity of Porous Syetems," Int. J. Heat
Mass Transfer, Vol. 11, pp l17- lb (la).
S. ksamune and J. M. Smith, '(Therm1 Conductivity of Beds of
Spherical Particles," I & EC Fun&mentals, Vol. 2, No. 2
(1%3), pp 136.142.
S. C. Cheng and R. I. Vachon, "A Technique for Predicting the
Them1 Con- ductivity of Suspensions, Emulsions and Porous
M~terliils," Int . J. Heat MRSS Transfer, Vol. 13, pp 537-546
(1WO).
I 1 "Investigation of Methods for Transpiration Cooling Liquid
Rocket Chanibers Pratt 8s Whitmy A i r c r a f t , WA FR-3390
(1969)
V I J. D. McClelland, "Materials and Structures Physical
Measurements Program, Report N&r TDW-930 (2240-64) TR-1,
Aerospace Corporation (1962).
J. C. Y. K o h , E. P. d e l Caeal, R. W. Evans, and V .
Deriugin, "Fluid Floir and Heat Transfer in High Temperature Porous
&trices for Transpiration Coolin$," rnLTR-66-70 (1966).
J. E. bans, Jr., "Thermal Conductivity of 14 Metals and A l l o
y s Up to llOO~,' IVACA RM E50LO7 (1951)
R. W. Pawell , "Correlation of Metallic Thermal and Electrical
Conductivities for Both %lid and Liquid Phases, ?* Int, J. liest
Mgss Transfer, Vol. 8, p~ 1033-1045 (1965)-
12
-
a()’ al
b
L
n
T
Constants, equation (7)
Constant in Wiederimnn-Franz-Lorenz equation
Constants, equation (16)
Constant in Wiedemnn-Franz-Lorenz equation
Constant, equation (8)
Temperature
o(
B x Them1 conductivity f Electrical reslstivfty
Tmpemture coefficient for conductivity, equation (5)
Temperature coefficient for resistivity, equation (6)
e Porosity Subscripts
0 Solid nraterial ( f = 0 )
t c
Iattice component
Electronic component
-
TABLE I DETAILS OF TES!BD SAMPLES
No .
1
2
3 4
5
6
7
8
9
10
11
l2
13
14
15
16
17
18
19
20
21
22
23
24
25
Miterials
304L SS Rigimesh
304L SS Rigimesh
304L SS Rigimesh
304L SS Rigimesh
304L SS Riginresh
304L SS Rigimesh
304L SS Iiigimesh
304L SS Rigimesh
304L SS Rigimsh
304L SS Sintered Powder
304L SS Sintered Powder
304L SS Sintered Powder
304L SS Sintered Powder
304L SS Sintered Potrtder
304L SS Sintered Powder
304L SS Sintered P d e r
304L SS Sintered Powder
Porosities
093
.093
093 . 203 . 3113 .203
385
385
385
0
0
.p2
0328
.2138
.2158
. 310 3 w
OFBC Copper Sintered Powder
OFHC Copper Sintered Puwder
WE Copper Sintered Pmder
CWHC Copper Sintered Puwder
OllrElC Capper Sintered Powder
llJFRC Copper Sintered Puwder
OFHC Copper Sintered Powder
WHC Copper Sintered Powder
0
0
. 1031
. 1031
.eo96
.2096
3009
3077
Size m
63 . 9x63 . 8~20.6 6.36 x 6.26~20.11
6 .& . 30~59.46 64.4Ak. 6x26 . 19 6 . 55x6 . 26x26 . 32
6.70~6 . 19~58.71 62 .75x62.594.46
6.64~6.45~25 . 52 6 . 73x8 . 60~59.07 25.5(d x 25.4 (1)
6.45 (a) x 31.5 (1)
25.6 (d) x 26.3 (1)
6.39 (a) x 12.4 (1)
25.6 (d) x 25.4 (1)
6.40 (a) x 26.6 (I)
25.4 (d) x 25.2 (1)
6.30 (a) x 40.1 (1)
25.4 (d) x 25.4 (1)
6.35 (a) x 31.1 (1)
25.1 (a) x 25.k (1)
6.35 (a) x 42.2 (1)
25.4 (d) x 25.4 (1)
25.4 (d) x 25.4 (1)
25.5 (a) x 25.4 (1)
6.27 (a) x 38.9 (1)
x F l m I or IJ or to Weave P Pattern A
P P A
P P x
P P A
P A
P h
P > P x P x
P x
P
x
P
1'
-
x :
H Elcu m a
H
6
!!I
I m m m
*
-
X 5 c crOM
Q Q , H
4
-
Stainless
3iginresh
- stainless
Pawbexi
' BFanze
Powders
(Reference 1)
-093
-203
385
0 - .$a
100 - lo00
loo- 700
d P -000578
.0005s?
-00333
-00345
0-5
-00309
-
0 0 *I
0 0
I 0 0
6
-
I
20
-
b
B I5 P 9
8 I n n 9
21
-
. . . . .
. . . . ....
. . . .
n
w d
m h .. .+a . . . . . .
. 4
..... ....
...................
. . . ... .-4, , . . . . . .
-* - . . . .
0
.. . . o l 2oQ L o a ) x200
' Mgiseeh L SOJsd &- -.- . . ! . !ltm@era%e (XI . . . . . .
. . .._.. . . . .
sl;atsrard p w e r .. ! . ' ... - m... . . . . . . . . . . i . '
:I:'. . . . . . . . . .'.- . . . . . . .
i " . ' . , I t . . . .......... . . . . . . . . . . . . . . . .
. . I f . . .. 1 , - - , . : . , .
22
-
23
-
. . . . . . . . * . ,
. . . t , . . . . . . . . . . . . . . . . . . . . . . . . . - -
. . j 1 . 4
, a .. 1. ..................... ... !.. ... :.. .... :.' ....;.
;. .&&& lu: . . . . . ........ bt ' . 1 . . . . . . . .
. . . . . t : : : : i . . ; : . ;.+i; . . . . . . t .
---I-..-
. . . . - . . . . . -..--'i-ILiL . . . . . . .
. . . . . . . . . . . . .
24
-
. . . . . . . . . . . . . . . . . . .
.......... i .......... .............-.... . . . . . . . . . . .
. . . , , ..;a: , . :.:: + . . , , : .;*
..... ................ C . . I
, , .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . :
...... I . ..A .....:. :I .. : .
, , : ! . : I . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
. . ...................... I . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . ...-.-.... I
.......,...... . . . . . . . . . . . . . . . . . . . . . . . . .
..‘.. ................ . . . . : ! *
. . . . . . . . . . . .
-
. . . . . . .
..-. . . ... . ._. . . . . . . . . . .. . . . . . . . . .
, . . .
. . . . . . . . .. . .
.._.
. .
..
. .
...
..
.. .
26
-
37
-
* . . . .
. . . . . . . , ,
.. .. .. ..._ -.
. . . * . .).. .
-
. . . . . . . . . . . . .
.......... *... ---...--
...........I. .. .. .! i.:: j . . .. . . . . .
a
-
. . . . 7 : . . . ..
.....
. . .
...
. .:
... . .
. .
. .
....
.....
. -. . ..
.. t ...... -. . . .
. . . . . . .
. ........
...
... ..
....
. . . . .
........
. . . . . . . .. . . . . . _ . _ . . . . . . . . . . . . . . . .
. . . , . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 4 160 . x lo8 '100 : '320 . Bai . . . . . . . . . . . . . .
. . . * . . -1 . . . .. . . . . . . . :T'@ &&&.3.:
...... .-: ......
.......... .... ---_...- . . . . - . . . . . . . . . . . . . . .
. . . . . . . . . '-T- . . . . . . - . . . . . . . . . . . . . . .
. . . . . .
-.. 9, .... _-__. ....._-*.--. ._- . . . . .
. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .-
...-
. . . . . . . . , . : i . . . . . . . A .... i .... i . I . . .
. . , i . : . . . . I . . . . . . . . . . . . . .
-
. . . t . . .
. . . . I _ _ . . . , . . ---. . . . . .
. . . . . . . . . , . . . . -___. . . . , .
, . . , . . .
-
. . . . . ....................... - . . . .. . . . . . - . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! _ :
. .
. . ................ . . . . . . . j . . . . . . . . . . . . . .
. .
j . pJ . . ~. ....... .* ....... ! : ..; . . . . . . . . . . . .
.......... ,. . . . * . . . . . . . . . .- ....... . . i .........
i... . . . .
. . . . . :... ....... .-*. 21 . . . . . . .......... _ . . . -
. .. . . . .
. . . . . . . . . . . . .
: : .# ... ...--.- .. . . . . . . . . . . .
. . . .-.----... .-. _.-. .n . . . . . . . ....A . . .
.......... .t-.-.-.
. . . . . . -*. -... ... . . . . . . . . ..::: ' c;;t ' % d
:
. . . . . . . . ......... r-. :. . . : :x:g::.. :---... .. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . L ....
--.-.--. ...
. . . . . . . . ?'. ........ -..... I 4 1. . : . i '.: . . . . .
. . . . . .
! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . L . . . . . . . . . . . . . . . . i . .
. . . .......... ,. . f..-;--- . . . . . . . . . . . :.:':: .
:i.:..:::f +-- . . . . . . . . . . . . . . . . . . . . . . . .
................... _. . . . . . . i .... . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
. . . . . . . . .
........... 9 $ : . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .._-. . .
. . . . . ./ . . . . .:
I . .
. . . . . . . . . . . . . . . . - . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . ... ....... ............................ ...... . . .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . ................................ . . . . . . . . . .
.
i J
.- -. . .
. .
. .
. . . ..__...._
............................. . . . . . . . . . . . . . . .. -.
. . . --. ..................................... -.. ... ~.- y.'- .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
x- . < . . . . . . ............................. : . . . . .
. . . . . p.d . . . . . . . . . . ... -.-. - ... *--. -. I..
......... . . . . . . . . . . . _- ._ . . . . . . . . . . . . . .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . . . . t t y . . . . . ; .... I . . . . . . .
:.. ..... -:. ............. : . .., . . . . . . . . . . . . . .
. . . . ~ . . . . . i . . . . f . . . . ;
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . . . . . .
. . . . . .
..T1 . . ::.. ~ : . t : : . . I : '.: . . . . . .
......_........_.... ................................... . . . .
. . . . . . . . . . . r.-.. . . . . . . . . , . . . . . . . . . . .
. , . . . .-.& _._____-... . ___..-. ................. . . . .
. . . . . . . . . . . . . .. ...---- . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .) . . . . . . . : . . . . .
. _ _ ............................. . . .
. . _ . . . . . . . . . ......................... . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . ................................................... ..
. . . . . . . . . . .. 1 . . . . . -.. ...--.--...-.-..-.-... -...-
.......... ............... . . . . . . . . . . . . . . . . . . .
.
................ * . ................................. . .
. : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . _. .. - ................. _. -. . . . . . . . . . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . -.. -.
.-. . . . . . . . . . . . . . .
.........................
..
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . i . . . . . . . . .
. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . :. . . ! . . . . . . . . . ...................
................. ..... . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . i ' . -.__--- --- -- ..-.---.-... r
......-............-. -.--__.. ....... " ._ k '::" ::::.. ~ ,:.:; I
. ::: : : . : : . :
. , .. . . . . . ! ::,. .:. : . : : . I . . : . . : . I . : ,.:
..::, ::; ::: .: :. . .
. . ._ _. ' . i '.: . :. ' .. ' I : : . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . ............ ............ . . .... . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
..' 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
1 -_. .-...... ---. -_. .^_._ -... A... ...... .*. .i.. .*. .-.
i .........- L.. ------.---...--i-. i..- -...--.-1- --......
-..--.--- .....-. - ........................................
.
-
. . .
.. . . . . . . . . , PÎ- . . . . . . . . . . . .
. . . ... . . . . -..-I..--.- . . . . . . . . . .
. . . . . . . . . . . . . ... .
33
-
APPENDIXA - I V I M R I C A L W L l 3 The computational
proce?z-es outlined 1% Section 4.0 are demonstrated i n the
following example where the conducitvfty of 3ObL stainless steel
powders at 500% having a porosity of 0.31 is 30 be compiited
fro& different k n m inform- tion.
(1) Given LO = 21.75 w tn -1 c-1 a t T = ~ O C
From equation (lQ), the conductivity at 500°C and
9 = o
{ = .3l is
= 21.75 (.35186) = 7.65 1 - .31 x = 21.75 1 + 10 (.3112
(2) Given A,, = 16.3 Wmol CD1 at T = 100% e From equation ( 5 )
,
= 0
= 0 is X et soooe and f
1 - 0.31 x 122.1 = 7.78 1 + 10 (.31)2
(3) Siven fo - 104.5 x ~ r o m equation (191,
a he = 2.16 x 10-
From equation (lo),
.
500 + 273 + 5.956 = 21.9
104.5 x loo8 the thernml conhctivity at ?C?(?% and 5 = .3l
is
X = 21.9 = 7.71 1 4 10 (.31!2
From equa+,ion
A@= 2.16 x
(lg), the therm1 conductivity at lW0C and f = 0 is
4. 5.H = 16.18 w m - 1 c-l
34
-
0 From equation ( 5 ) , the thermal conductivity at 500 C and 5
= 0 I s
= 16.18 [l + -00089 ( b o ) ] = 21.94 Wm -1 f1 A,
From equation (lo), the t h e m 1 conductivity at 500% and f =
.3l is
1 - .31 1 = 21.94 = 7.72 1 + 10 (.31)2
Measured value of them1 conductivity at w o C and .3l porosity
as given by Table I1 (b) is 7.6 Wm'l Col. Thus, the maximum
deviation between the computed and measured results I s 2.4s.
35